discussion

discussion_1

Beware: Under construction!

In the interest of speed rather than elegance I'm throwing this together quickly!

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Part I: Misalignments and Acceptance

2/7/01 Here I'll summarize what led to my remarks at the Hall A Collaboration meeting last month (January). (The talk.)

I received the following plots from Feng Xiong (maple@jlab.org)

What is shown is, for 2 different beam energies and scattering angles, the distribution of trajectories at the dipole exit for elastic scattering off ³He. In both cases the effective target length was more than the spectrometer acceptance. The "experiment" plots are for trajectories in the real data reconstructed at the target and then run through the spectrometer to the dipole exit as in the simulation. The "simulation" plots are the same distribution in a pure simulation. (I think the simulation used MCEEP). Notice that the real trajectories don't fill the dipole exit as indicated by the red boxes. Feng found that the elastic peak in his simulation was much larger than in his data. However, if he went back and redid the simulation using the reduced asymmetric aperture at the dipole exit suggested by the plots above he got pretty good agreement between the data and the simulation, at least in the elastic peak region. (See the plot below)

Remarks from Feng re the above: "Actually after I performed narrower cuts on Q3 exit as we discussed before and also some cuts on Q3 entrance, the 1 pass simulation looks much better than before(though not perfect). I think this is the best I can achieve after many trys. More important, I find that the aperture cuts I made on 1 pass and 2 pass are actually very similar to each other,this is a very important self-consistency check on this method."

Since this effect is asymmetric and quite large I suspect that what we are seeing is the result of misalignments in the spectrometer causing a reduced "effective aperture" at the dipole exit. Just adjusting the the quadrupole fields in the spectrometer model wouldn't produce the asymmetry and to change the envelope on the order of several (~2.5) cm would require a very large change in the quadrupole fields, which we must rule out on the basis of the field measurements (The alternative is to tell the QMM collaboration that they don't have a clue how to measure a magnet and I don't believe that for even a minute). So, I went through an exercise, which I outlined in some detail in my talk to see if I could reproduce the effect using simple misalignments of Q₁ and Q₂ and I indeed succeeded in that effort.

What I propose at this point is that the various experiments and people interested in the acceptance issue in general repeat Feng's exercise and determine for your data a set of "effective apertures" and report them to me/the HRS optics working group aka. halla_hrs_optics@jlab.org -JJL (lerose@jlab.org)

Part II: Attempts to Improve the Spectrometer Model

Re Transfer matrices and databases
R₁₆	(x\|d) = 1/D₁₀₀₀
R₂₂	(q\|q₀) = 1/T₁₀₀₀
R₁₂	(x\|q₀) = -D₀₁₀₀/(D₁₀₀₀ * T₀₁₀₀)
R₃₃	(y\|y₀) = +P₀₀₀₁
R₃₄	(y\|f₀) = -Y₀₀₀₁
R₄₃	(f\|y₀) = -P₀₀₁₀
R₄₄	(f\|f₀) = +Y₀₀₁₀

The Problem: The database (Nilanga's latest best) gives a set of 1^st order transfer matrix coefficients that, particularly for the transverse terms (y and f), doesn't agree with what one calculates using the known values of the various magnet parameters (product of field gradient and length for the quadrupoles, entrance and exit shim angles for the dipole and field index for the dipole. This is summarized for 1^st Order TRANSPORT in the table below. This discussion refers to the left arm.

	Jan. '01 database	TRANSPORT	D
x\|d	11.7979	11.91013	1.0%
x\|q	-0.00949	-0.03853	306.0%
q\|q	-0.4353	-0.49733	14.2%
y\|y	-0.7094	-0.62406	-12.0%
y\|f	-0.07698	-0.08262	7.3%
f\|y	3.425	3.80235	11.0%
f\|f	-1.126	-1.09902	-2.4%

Some remarks:

1^st Order TRANSPORT should do "pretty well" and it does

The disagreement for x|q although large when viewed as a percentage of the actual term is not really important in that the term is very small anyway

The terms in the database are the 1^st order terms from independent fits to data. There may be some trading back and forth of strength between 1^st order and 3^rd order terms that are allowable within the constraints of the fit and give the same answer when all terms are used to make a reconstruction. However, they may not be the "true" 1^st order terms in the sense that TRANSPORT evaluates them. As evidence the determinant of the transverse matrix is 1.06. It should be 1.

The problem in TRANSPORT seems relatively simple. There are six magnet parameters that one can adjust and seven matrix elements to fix. One picks six matrix elements, demands that those matrix elements have the values in the Jan. '01 database and lets TRANSPORT find a solution. Given that the Jan '01 transverse matrix determinant is not equal to one forces you to choose one of the transverse matrix elements as free from constraint. I did this and ran TRANSPORT four times. My hope was that I would get some collection of small adjustments to the assumed values of the magnet parameters that would give me a good match to the database optics. That isn't what happened. Except for the case where the (y|y₀) term was allowed to vary freely, I got back magnet parameters that were manifestly impossible. In that exceptional case the magnets still varied outside what would be a reasonable range based on the field measurement results. The results are summarized in the plot below, which shows the percent variation in the magnet parameters and the associated relative difference between the database matrix elements and those in the TRANSPORT solution.

Observations:

Getting the 1^st order matrix right doesn't guarantee that you'll get the acceptance right. The solutions found in trials 1', 1'', & 1''' above would give very different acceptances than the actual spectrometer.
The problem is almost certainly over-constrained given that there is probably some latitude in the actual true values for the 1^st order matrix elements.

Next step: Repeat the process but leave 2 terms (f|y₀ & f|f₀) in the transverse matrix free to vary. In this case I chose to fix the field index in the dipole at -1.24, -1.25, and -1.26 (a reasonable range of values). The index is pretty well known and couples only very weakly to the y-terms, which are where the problem is. The results follow:

Observations:

Here at least things are reasonably well behaved.
Notice that the choice of field index makes very little difference.
The problem remains that the variations in the quadrupoles are large compared to the uncertainties in the field measurements.

Next Step: Move on to Raytracing in SNAKE.

Discouraged but refusing to give up, I went on and applied the last TRANSPORT strategy using my SNAKE model for the spectrometer. The reader should bear in mind that SNAKE does not explicitly evaluate matrix elements, it merely traces trajectories. What I do is run a sample of 11 trajectories through the spectrometer involving small systematic variations in the target parameters (x, q, y, f, d) and use them to infer the 1^st order matrix elements. To mimic the TRANSPORT strategy an iterative process is used (it's slow, painful, and boring, but it gets you there eventually).

First, you establish the 1^st order matrix for a given configuration of the magnets. In this case the configuration I call comm_8 is my best estimate of what the magnet configuration in the left arm (formerly electron arm) based on the parameters of the tune and the results of the field measuring. Here it's worth noting that instead of using maps from the field measurements, I use the effective lengths and field strengths as measured as input to an analytic routine (stolen from RAYTRACE) that describes multipolar magnets in terms of their lengths and field strengths at a fixed radius in the uniform field region with a parametrized description of the fringe field fall off.

Second, you make several runs wherein you systematically vary each of the relevant magnet parameters one at a time by a small amount. You collect a set of 1^st order matrices and use them to determine how each parameter couples to a change in the matrix elements.

Third, you determine from that set of couplings a set of changes in magnet parameters that produces the desired change in the 1^st order matrix. Then in principle you apply those changes and get a spectrometer with the desired 1^st order properties. The trouble is life is not quite that simple. Changing a magnet changes to some extent the coupling of subsequent magnets to the matrix elements. So, you must invoke Ingvar Blomqvist's one half rule; you change each magnet by one half what you calculate you should. Establish a new matrix and repeat the process. In this way after a few iterations, if you're lucky, you converge on a configuration with the desired matrix. The results of my effort so far in this case follows:

	Jan'01 db	Comm_8	D	Comm_9	D
x\|d	11.80	11.91	0.9%	11.6875	-0.9%
x\|q	-0.09	-0.20	109.0%	-0.1379	45.3%
q\|q	-0.44	-0.47	7.7%	-0.442	1.5%
y\|y	-0.71	-0.60	-16.1%	-0.6856	-3.4%
y\|f	-0.77	-1.27	65.5%	-0.7506	-2.5%
f\|y	0.34	0.36	6.1%	0.16634	-51.4%
f\|f	-1.13	-0.90	-19.9%	-1.2768	13.4%


	Q1	0.2445		0.24815	1.5%
	Q2	-0.1939		-0.1873	-3.4%
	rot1	-30		-31.275	4.3%
	n	-1.263		-1.263	0.0%
	rot2	-30		-30.691	2.3%
	Q3	-0.1794		-0.1661	-7.4%

"Jan'01 db" is Nilanga's database. "Comm_8" is my "standard" spectrometer model and "Comm_9" is the "new and improved" version. The D's are relative differences between the database and the calculated matrix elements and the relative differences in the magnet settings between Comm_8 and Comm_9. (Remember Comm_8 is what's supposed to be there if you implicitly believe the field mapping.)

Observations:

The process appeared to be converging when I stopped. With another iteration or two I might have been able to make the matrix elements agree a little better but the gross features of the magnet settings would not have changed.

Previous comments about (x|q₀) apply here as well.

The changes in the quadrupoles are still out of bounds.

There are uncertainties in the matrix elements themselves, which could be driving the unacceptable magnet values.

Using an analytic model for the quadrupole fields instead of actual maps should not in any significant way effect the 1^st order properties discerned from the model (i.e. Comm_8 or Comm_9). In the interest of leaving no stone unturned we can check this.

What's next?

Look carefully at the database and how it was formed. Try to get a feeling for how accurately the matrix elements are determined. See if there are any large 3^rd order terms. (Nilanga and I should do this)
Try the above procedure using maps instead of analytic quadrupoles (Hélène, Stephanie, Gilles???)
…

Would be spectrometer optics working group members, please feel free to make comments, ask questions, do some work!

In the interest of not generating too much e-mail for everyone, I suggest that if you have specific questions about the work I've outlined here send them to me (lerose@jlab.org). I'll answer to the whole mailing list, with the original questions. Naturally, comments and discussion for the whole mailing list can be sent directly to halla_hrs_optics@jlab.org .

-JJL

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